Orthogonal Projections Prepared by Vince Zaccone For Campus
Orthogonal Projections Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB
Orthogonal Projection For a vector u in ℝn we would like to decompose any other vector in ℝn into the sum of two vectors, one a multiple of u, and the other orthogonal to u. That is, we wish to write: Where for some scalar α, and z is a vector orthogonal to u. Here is a formula. To get this, just set u • z=0 and rearrange. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB
Orthogonal Projection For a vector u in ℝn we would like to decompose any other vector in ℝn into the sum of two vectors, one a multiple of u, and the other orthogonal to u. That is, we wish to write: Where for some scalar α, and z is a vector orthogonal to u. Another version of the formula. This one shows the unit vectors in the direction of u. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB
Orthogonal Projection Here is an example using vectors in ℝ 2. Find the orthogonal projection of y onto u. Subtracting yields the vector z, which is orthogonal to u. Check this by finding that z • u=0. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB
Orthogonal Projection One important consequence of the previous calculation is that we have manufactured an orthogonal basis for the vector space ℝ 2. This idea can be very useful in a variety of situations. The original set of vectors was a basis for ℝ 2, but the vectors were not orthogonal. To find an orthogonal basis, we simply found the projection of one vector onto the other, then subtracted it, leaving an orthogonal vector. We can go a step further and find an orthonormal basis by simply dividing each vector by its magnitude. Original basis Orthogonal basis Orthonormal basis In this example, we only needed two basis vectors (ℝ 2 is 2 -dimensional), but if we are dealing with a larger space this process can be repeated to find as many vectors as necessary. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB
Orthogonal Projection The process of constructing an orthonormal basis in the way we have described is called the Gram-Schmidt process. Here is another example, this time with vectors in ℝ 3. The given set of vectors form a basis for ℝ 3. Use the Gram-Schmidt process to find an orthonormal basis for ℝ 3. Step 1 is to find the projection of v 2 onto v 1, and subtract it from v 2, leaving a new vector that is orthogonal to v 1. I will call this new vector v 2*. Can scale vector to avoid fractions Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB
Orthogonal Projection We now have two orthogonal vectors. To manufacture third one we will project v 3 onto both of these, and subtract those projections to obtain a vector that is orthogonal to both v 1 and v 2*. Call this one v 3*. We can scale these vectors however we want, so for convenience we can use the following orthogonal set: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB
Orthogonal Projection The last step is to normalize the vectors. Simply find the length of each vector and divide, obtaining a vector of length 1 that points the same direction. Here is an orthonormal set of vectors that spans ℝ 3. If we make a matrix with these vectors as columns we get a very convenient orthogonal (orthonormal) matrix with the property that U T=U-1 Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB
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